congratulations, all your steps were perfect, however you missed the point.

you have an equation to solve for two variables, however you only have one equation. consider:

we cannot solve for A and B as exact numbers, for instance whatever A is, we can find a value for B so that the equality is true:

let A = 2
then
2B = -A
2B = -2
B = -1

or if we let A be 7
2B = -7
B = -3.5

so there are infinitely many solutions that satisfy A + 2B = 0
likewise there are infinitely many solutions which satisfy 6A - 7C = 9
you had A = (9+7C)/6
let C = 3 then
A = (9+7*3)/6 = 30/6 = 5
so if C = 3, then A = 5
or if C = -9
A = (9 + 7*(-9))/6 = (-54)/6 = -9
so if C = -9, A = -9.
so you cannot solve for one unique solution of A and C, only a relationship between the two, so that whatever let one variable be, you can find a solution for the other one.